A signature of chaotic behavior in dynamical systems is sensitive dependence on initial conditions, and Lyapunov exponents measure the rates at which nearby orbits diverge. One might expect that geometric expansion or stretching in a map would lead to positive Lyapunov exponents. This, however, is very difficult to prove - except for maps with invariant cones (or a priori separation of expanding and contracting directions).
Consider a sequence of random walks on $\mathbb{Z}/p\mathbb{Z}$ with symmetric generating sets $A= A(p)$. I will describe known and new results regarding the mixing time and cut-off. For instance, if the sequence $|A(p)|$ is bounded then the cut-off phenomenon does not occur, and more precisely I give a lower bound on the size of the cut-off window in terms of $|A(p)|$. A natural conjecture from random walk on a graph is that the total variation mixing time is bounded by maximum degree times diameter squared.
Title: Equidistribution of expanding translates of curves in homogeneous spaces and Diophantine approximation.
Abstract:
We consider an analytic curve $\varphi: I \rightarrow \mathbb{M}(n\times m, \mathbb{R}) \hookrightarrow \mathrm{SL}(n+m, \mathbb{R})$ and embed it into some homogeneous space $G/\Gamma$, and translate it via some diagonal flow
Manchester building, Hebrew University of Jerusalem, (Room 209)
Title: Indistinguishability of trees in uniform spanning forests
Abstract:
The uniform spanning forest (USF) of an infinite connected graph G is the weak limit of the uniform spanning tree measure taken on exhausting finite subgraphs of G. It is easy to see that it is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle ('91) asserts that when G=Zd, the USF is almost surely a connected tree if and only if d=1,2,3,4.
Manchester building, Hebrew University of Jerusalem, (Room 209)
Title: Positive entropy actions of countable groups factor onto Bernoulli shifts
Abstract: I will prove that if a free ergodic action of a countable group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countable groups the well-known Sinai factor theorem from classical entropy theory. As an application, I will show that for a large
class of non-amenable groups, every positive entropy free ergodic action satisfies the measurable von Neumann conjecture.
We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. We present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity, as well as characterize the growth of the variance of N(T).
For a given deterministic measure we construct a random measure on the Brownian path that has expectation the given measure. For the construction we introduce the concept of weak convergence of random measures in probability. The machinery can be extended to more general sets than Brownian path.
Equilibrium states of affine iterated function systems
Motivated by the long-standing problem of finding sharp lower estimates for the Hausdorff dimension of self-affine sets, I will describe some recent results on the equilibrium states of the singular value function. These equilibrium states arise as candidates for the measures of maximal Hausdorff dimension on self-affine sets. In particular I will discuss a sufficient condition for uniqueness of the equilibrium state (from joint work with Antti Käenmäki) and an unconditional bound for the number of ergodic equilibrium states (from joint work with Jairo Bochi).
In his influential disjointness paper, H. Furstenberg proved that weakly-mixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern
research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this
disjointness theorem for the case of the horocyclic flow on a compact Riemann surface.
If N denotes a Poisson process, a splitting of N is formed by two point processes N_1 and N_2 such that N=N_1+N_2.
If N_1 and N_2 are independent Poisson processes then the splitting is said to be Poisson and such a splitting is always available (We allow the possibility to enlarge the ambient probability space).
In general, a splitting is not Poisson but the situation changes if we require that the distributions of the point processes involved are left invariant by a common underlying map that acts at the level of each point of the processes.
A sequence of Picard/Galois orbits of special points in a product of arbitrary many modular curves is conjectured to equidistribute in the product space as long as it escapes any closed orbit of an intermediate subgroup. This conjecture encompasses several well-known results and conjectures, including Duke's Theorem, the Michel-Venkatesh mixing conjecture and the equidistribution strengthening of André-Oort in this setting.
